ESTADÍSTICA (2009), 61, 176 y 177, pp. 7-16
© Instituto Interamericano de Estadística

ESTADISTICA (2009), 61, 176 y 177, pp.
c Instituto Interamericano de Estadı́stica

TESTING A HYPOTHESIS ABOUT A STRUCTURAL
COEFFICIENT IN A SIMULTANEOUS EQUATIONMODEL FOR KNOWN COVARIANCE MATRIXTHEODORE W. ANDERSON
Department of Statistics and Department of Economics, Stanford University
twa@stanford.edu
NAOTO KUNITOMO
Graduate School of Economics, University of Tokyo
kunitomo@e.u-tokyo.ac.jp
ABSTRACT
There is now considerable econometrics literature on estimating the coefficientsof one equation in a simultaneous equation model. Recently, econometricians areentertaining the problem of testing a hypothesis concerning the coefficients of asingle equation. In this paper we derive the likelihood ratio test for a vector ofcoefficients in the case when the error covariance is known.
Key words
Error covariance matrix; likelihood ratio test; simultaneous equation model; testing hypothesis.
RESUMEN
Existe considerable literatura en econometrı́a sobre la estimación de los coeficientes de una ecuación individual en un modelo de ecuaciones simultáneas. Recientemente, algunos econometristas se están ocupando del problema de la prueba
de una hipótesis concerniente a los coeficientes de una ecuación individual. En
este artı̈culo deducimos una prueba de razón de verosimilitud para un vector decoeficientes en el caso en que la matriz de covarianzas de errores es conocida.
Palabras clave
Matriz de covarianzas de errores; prueba de razón de verosimilitud; modelo deecuaciones simultáneas; prueba de hipótesis.

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21. Introduction
There is a considerable econometrics literature about estimating coefficients of
one equation in a simultaneous equation model (Anderson and Rubin (1949), forexample). More recently, econometicians have been studying the problem of testing a hypothesis concerning the coefficients of a single equation. See Andrews,
Moreira and Stock (2006), for example. In a previous study Anderson and Kunitomo (2007) have derived the likelihood ratio test of a vector of coefficients when it
is assumed that the single equation is identified and the covariance is estimated.
In the present paper we derive the likelihood ratio test in the case of the error
covariance matrix known.
2. The model
Let Y be a T × G matrix of endogenous or dependent variables; let Z be a T × K
matrix of exogenous or independent variables; and let V be a T × G matrix ofunobservable errors or shocks, which are statistically independent of Z. The rowsof V are assumed independent, the t-th row of which has mean E(vt ) = 0 and0covariance E(vt vt ) = Ω. (The matrix Ω is assumed known.) The ”reduced form”
of the model is
Y = ZΠ + V,
(1)
where Π is a K × G matrix of parameters. Statisticians will recognize (1) as amultivariate regression.
A single structural equation of interest may be written as
Yβ = Zγ + u,(2)

where u = Vβ. To distinguish this equation from other linear combinationsof Y and Z, some ”identifying” assumptions concerning β and γ are made; inparticular some components of β and γ may be assumed 0. The relation between(1) and (2) impliesΠβ = γ.
(3)
The equation (1) is said to be identified by 00 s if given Π (3) has a unique solution
00for (β , γ ) except a constant of proportionality.
In this paper first we pay special attention to the case that γ = 0. Then therelation (3) isΠβ = 0.
(4)
Then β is identified if Π has rank G − 1.
We consider the null hypothesis H0 : β = β 0 , where β 0 is a specified vector. Thisnull hypothesis is equivalent to the hypothesis that Πβ 0 = 0.

9ANDERSON AND KUNITOMO: Testing a hypothesis about a structural coefficient...

3The unrestricted regression coefficient Π can be estimated by the sample regression00
P = (Z Z)−1 Z Y.
(5)When Ω is known and V is normal, the matrix P constitutes a sufficient set ofstatistics. The density of P for Ω = IG is
10− 21 T G
L(Π) = (2π)
exp − tr(P − Π) A(P − Π) ,(6)2
where0A = Z Z.
The likelihood maximized with respect to Π is1

LH2 = (2π)− 2 T G ,where H2 denotes the model with no restriction on Π.
The model and the inference problem have some invariance properties. If Z isreplaced by Z̃ = ZC and Π is replaced by Π̃ = C−1 Π, then the model (1)is unchanged since Πβ = 0 is transformed to Π̃β = 0 The null hypothesis isunaffected. Note that C is an arbitrary nonsingular matrix.
0If Y is replaced by YΦ = Y∗ , V is replaced by VΦ = V∗ (vt∗ = Φ vt ), Πis replaced by ΠΦ = Π∗ , β is replaced by Φ−1 β = β ∗ , and β 0 is replaced by0
Φ−1 β 0 = β ∗0 , the model is unchanged with Ω replaced by Φ ΩΦ = Ω∗ . When0Ω = Ω∗ = IG is required, Φ Φ = IG , that is, Φ is an orthogonal matrix.
3. Likelihood Ratio Tests
We denote the model with Π having rank G − 1 as H1 . We shall derive thelikelihood ratio criterion to test H0 when H1 is assumed, namely LH0 /LH1 , as theratio of the criterion to test H0 vs H2 and the criterion to test H1 vs H2 .
LH0 /LH2
L H0=.
L H1
LH1 /LH2
The hypothesis that Πβ 0 = 0 can be considered as the hypothesis that Π hasrank G − 1 and Πβ 0 = 0. When G = 2, Π has rank 1, that is, can be written as00µγ for the K-vector µ and the 2 vector γ, and that γ β 0 = 0.
0Lemma: If G = 2 and Π = µγ , then the likelihood function of P maximized
with respect to µ is1
1 0 000−Tmax L(µγ ) = (2π) exp − trP AP + γ P APγ .
µ2210ESTADÍSTICA (2009), 61, 176 y 177, pp. 7-164Proof:10 00L(µγ ) = (2π) exp − tr(P − µγ ) A(P − µγ )2i
1 h000000−T= (2π) exp − tr PAP − γµ AP − P Aµγ + γµ Aµγ
20−T0

is maximized at µ = P γ/γ γ. Q.E.D.
To eliminate the indeterminacy of a scale factor in β we require0β β = 1.
An alternative normalization is0β Ψβ = 1,0where Ψ is a known positive semi-definite matrix. To specify β = (1, β 2 ), forexample, we take

1 0
Ψ=.
0 0When Π is K × 2, then Πβ = 0 implies
0β γ = 0.
0Let γ 0 be a vector such that β 0 γ 0 = 0. Then
11 0 00−TLH0 = (2π) exp − trPAP + γ 0 P APγ 022100−T= (2π) exp − β 0 P APβ 02since00000trPAP = β P APβ + γ P APγ.

The likelihood ratio criterion for testing H0 against H2 is
L H01 0 0= exp − β 0 P APβ 0 .
L H22When the null hypothesis H0 is true,−2 log

L H000= β 0 P APβ 0L H2ANDERSON AND KUNITOMO: Testing a hypothesis about a structural coefficient...

115
has a χ2 −distribution with K degrees of freedom since Pβ 0 has the distribution
N (0, σ 2 A−1 ).
Now consider
H1 : rankΠ = G − 1,0that is, Π is written as Π = µγ . The likelihood of P maximized subject to0
Π = µγ is0

LH1 = max max L(µγ )γ µ0011γ P APγ0−T= (2π) exp − trPAP + max
22 γγ0 γ()
001β P APβ−T= (2π) exp − min.
02 βββThe likelihood ratio criterion for testing H1 against H2 isL H1L H2(001β P APβ= exp − min02 βββ()0b P0 APβb1β= exp −,02bβbβ)

b is the maximum likelihood estimator of β for H1 .
where βWhen the null hypothesis H1 is true,
LH−2 log 1L H200β P APβ= min0βββ0b P0 APβbβ=0bβbβ

has a limiting χ2 -distribution with 1 degree of freedom (Anderson and Rubin,0001950). Note that minb (b P APb/b b) is the smaller root of0P AP − dI2 = 0.

12ESTADÍSTICA (2009), 61, 176 y 177, pp. 7-166The likelihood ratio criterion for testing H0 against H1 isL H0L H1LH0 /LH2
LH1 /LH2()001 0 01β P APβ= exp
β P APβ 0 + min02 02 βββ#)( " 0 0b P APβb β 0 P0 APβ1 β0− 0 0.
= exp
02ββbb0 0ββ=

Equivalently, the likelihood ratio criterion for testing H0 against H1 is0LH−2 log 0L H100b P0 APβbβ 0 P APβ 0 β−.
=00β0β0bβbβ(7)b The
Note that the criterion does not depend on the normalization of β 0 and of β.
criterion (7) has been proposed by Moreira (2003). He called (7) the conditional
likelihood ratio.
Corollary: The likelihood ratio criterion (7) is invariant with respect to thegroups of transformations
Z̃ = ZC, Π̃ = C−1 Πand00Y∗ = YΦ, P∗ = PΦ, β ∗ = Φ β, β ∗0 = Φ β 0 ,(8)0where ΦΦ = I. When Ω = σ 2 I, the transformation (8) is appropriate. A0normalization of β Ψβ = 1 is transformed to00β ∗ Φ ΨΦβ ∗ = 1.
0

When G = 2 and Ω = I, the condition β β = 1 implies that β can be written as
cos θβ=(9)sin θand γ can be written as
γ = constant ×− sin θcos θ.

ANDERSON AND KUNITOMO: Testing a hypothesis about a structural coefficient...

1370The term maxµ L(µγ ) is independent of the constant which can be taken as 1.
The null hypothesis β = β 0 wherecos θ0β0 =sin θ0implies that the null hypothesis β = β 0 is equivalent to θ = θ 0 and
− sin θ0γ = γ0 =.
cos θ0Anderson and Kunitomo (2007) derived the likelihood ratio criterion for testingH0 against H1 when Ω is unknown and must be estimated. In that case let00

H = Y Y − P AP.
The maximum likelihood estimators of Π and Ω are P and (1/T )H (under H2 ).
Then"
#
0000
L∗H0
β 0 P APβ 0b P APb−2 log ∗ = log 1 +− log 1 + min0bL H1b0 Hbβ 0 Hβ 0
is the likelihood ratio criterion for testing H0 against H1 . It might be observedthat Anderson and Rubin (1949) derived the likelihood ratio criterion for testingH1 against H2 explicitly and the LR criterion for testing H0 against H2 implicitly(the converse of a confidence region), but did not derive the LR criterion fortesting H0 against H1 .
pp

When the null hypothesis is true, Pβ 0 → 0, (1/T )H → Ω as T → ∞. Assume(1/T )A → A0 . Then expansion of log(1 + x) = x + op (1) for x → 0 yields
0000L∗b P APbβ P APβ 0−2 log ∗H0 = 0 0− min+ op (1).
bL H1b0 Hb
β 0 Hβ 0The limiting distribution of −2 log[LH0 /LH1 ] is the same as the limiting distribution of −2 log[L∗H0 /L∗H1 ]. Under the null hypothesis the limiting distribution is χ2with 1 degree of freedom.
4. More General Models
Suppose that Y, Z, β and γ in (2) are
Y = (Y1 , Y2 ), Z = (Z1 , Z2 ), β =β10

γ=γ10,

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ESTADÍSTICA (2009), 61, 176 y 177, pp. 7-16

8where Y1 and Y2 have G1 and G2 columns, respectively, and Z1 and Z2 have K1and K2 columns, respectively. Then (2) isY1 β 1 = Z1 γ 1 + u.
Let
Π=Then (2) and (10) implyΠ11 Π12
Π21 Π22Π11 β 1
Π21 β 1,Ω=





γ10=



Ω11 Ω12Ω21 Ω22(10)
.

(11)

and u = V1 β 1 , where V = (V1 , V2 ). The second subvector of (11) corresponds
to (4).
It will be convenient to transform the model so that the two sets of exogenousvariables are orthogonal. Let  000A11 A12Z1 Z1 Z1 Z2A=ZZ==00
A21 A22Z2 Z1 Z2 Z2and−1Z2.1 = Z2 − Z1 A−1
11 A12 , A22.1 = A22 − A21 A11 A12 .

Define (Π∗11 , Π∗12 ) = (IK1 , A−1
11 A12 )Π. ThenΠ11 β 1ZΠ = (Z1 , Z2.1 )= (Z1 , Z2.1 )Π∗ ,Π21 β 1by defining Π∗ . Define also
 0 


A11OZ1∗A =[Z1 , Z2.1 ] =,O A22.1Z2.1 0  ∗ 0P1Z1A−1∗∗ −111 Z1 Y.
P = (A )Y==−1P∗2Z2.1
A22.1 Z2.1 Y
The density of P∗ is defined from the density of P given (5) as

1∗∗ 0
∗
− 21 T G1∗∗
L(Π ) = (2π)
exp − tr(P1 − Π1 ) A11 (P1 − Π1 )2


1
10×(2π)− 2 T G2 exp − tr(P∗2 − Π∗2 ) A22.1 (P∗2 − Π∗2 ) ,2ANDERSON AND KUNITOMO: Testing a hypothesis about a structural coefficient...

159where Π∗1 = (Π∗11 , Π∗12 ) and Π∗2 = (Π21 , Π22 ). The maximum of the first part of1L(Π∗ ) occurs at Π∗1 = P∗1 and is (2π)− 2 T G1 . The maximum of the second partwith respect to Π22 is110L(Π21 ) = (2π)− 2 T G2 exp − tr(P21 − Π21 ) A22.1 (P21 − Π21 ) .
2This corresponds to (5).
5. Distribution of Test Criterion
The matrix

0G = P AP
has the noncentral Wishart distribution with T −K degrees of freedom, covariancematrix Ω = IG , and the noncentrality parameter0Γ = Π AΠ
when P has the density (6). The noncentral Wishart distribution was derived
by Anderson and Girshick (1944) with correction (1964) for Π of rank 1. The
criterion (7) can be written
00LHβ Gβb Gb−2 log 0 = 0 0 0 − min 0 .
bL H1bb
β0β0(12)Since the criterion is invariant with respect to orthogonal transformations (9), we
0
can write (12) for β 0 = (1, 0) and0

Π = µγ = µ(0, 1) = (0, µ)and0minbb Gb= d1 ,b0 b

the smaller root of20 = |G − dI2 | = d2 − (g11 + g22 )d + g11 g22 − g12.

Thus (12) isq122
g11 − g22 + (g11 − g22 ) + 4g12 .
g11 − d1 =2(13)The distribution of (13) can be found from the noncentral Wishart distribution.
See Anderson, Kunitomo, and Matsushita (2009).

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ESTADÍSTICA (2009), 61, 176 y 177, pp. 7-16

10References
ANDERSON, T.W. and KUNITOMO, N. (2007). “On Likelihood Ratio Testsof Structural Coefficients : Anderson-Rubin (1949) revisited,” Discussion Paper
CIRJE-F-499, Graduate School of Economics, University of Tokyo (http://www.e.utokyo.ac.jp/cirje/research/dp/2007).
ANDERSON, T.W., KUNITOMO, N., and MATSUSHITA, Y. (2009). “The Limited Information Maximum Likelihood Estimator as an Angle,” Discussion Paper
CIRJE-F-619, Graduate School of Economics, University of Tokyo (http://www.e.utokyo.ac.jp/cirje/research/dp/2009).
ANDERSON, T.W. and GIRSCHICK, M.A. (1944). “Some extensions of theWishart distribution,” Annals of Mathematical Statistics. 15: 345–357. (Corrections, 1964, 35: 923–925).
ANDERSON, T.W. and RUBIN, H. (1949). “Estimation of the Parameters ofa Single Equation in a Complete System of Stochastic Equations,” Annals ofMathematical Statistics. 20: 46–63.
ANDERSON, T.W. and RUBIN, H. (1950). “The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of StochasticEquation,” Annals of Mathematical Statistics. 21: 570–582.
ANDREWS, D.W.H., MOREIRA, M.J., and STOCK, J.H. (2006). “Optimal
Two-Sided Invariant Similar Tests for Instrumental Variables Regression,” Econometrica. 74: 715–752.
MOREIRA, M. (2003). “A Conditional Likelihood Ratio Test for StructuralModels,” Econometrica. 71: 1027–1048.

Received June 2009
Revised August 2009